A Multistep Legendre-Gauss Spectral Collocation Method for Nonlinear Volterra Integral Equations

We introduce a multistep Legendre--Gauss spectral collocation method for the nonlinear Volterra integral equations of the second kind. This method is easy to implement and possesses high order accuracy. In addition, it is very suitable for long time calculations. We also derive the optimal convergence of the $hp$-version of the multistep collocation method under the $L^2$-norm. Numerical experiments confirm the theoretical expectations.

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